About Multiplying Matrices:
In order to multiply two matrices together, the number of columns from the first matrix (leftmost) must be equal to the number of rows from the second matrix (rightmost). Once we have confirmed this fact, we set up a matrix for our product with the number of rows from the first matrix and the number of columns from the second matrix. To obtain a given individual entry, we use the row and column location of that entry. Once we have that information, we use the row to find the row to work within the first matrix (leftmost) and we use the column to find the column to work within the second matrix (rightmost). The given entry is found by obtaining the dot product of that given row by that given column. We can find all entries of the matrix using the same strategy.
Test Objectives
- Demonstrate the ability to multiply matrices
#1:
Instructions: Find AB.
$$a)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}-5 & 5 & -4\\ -2 & 2 & 2\end{array}\right]$$ $$B=\left[ \begin{array}{cc}-3 & -6\\ 2 & 3 \\ 6 & 1\end{array}\right]$$
$$b)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}4 & -12 & 1\\ 1 & 5 & 0\end{array}\right]$$ $$B=\left[ \begin{array}{ccc}-4 & 1 & 9\\ -3 & 7 & -6\end{array}\right]$$
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#2:
Instructions: Find AB.
$$a)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}-6 & -3 & -1\end{array}\right]$$ $$B=\left[ \begin{array}{ccc}-3 & 6\\ -2 & -6 \\ 6 & -1\end{array}\right]$$
$$b)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}9 & 2 & 7\\ 5 & 8 & -1 \\ 0 & 0 & -3\end{array}\right]$$ $$B=\left[ \begin{array}{ccc}-1 & 2 & 13\\ -3 & 0 & 7 \\ -4 & -1 & 12\end{array}\right]$$
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#3:
Instructions: Find AB.
$$a)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}-5 & 0 & 6\\ 5 & -1 & 1 \\ 2 & -1 & -6\end{array}\right]$$ $$B=\left[ \begin{array}{cc}0 & -1\\ -3 & -3 \\ -2 & -4\end{array}\right]$$
Instructions: Find ABC.
$$b)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}0 & -6 & 5\\ -1 & 2 & -1\end{array}\right]$$ $$B=\left[ \begin{array}{cc}-3 & -5\\ -5 & -1\\ -1 & -3\end{array}\right]$$ $$C=\left[ \begin{array}{ccc}0 & -1 & 2\\ 1 & -2 & 1\end{array}\right]$$
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#4:
Instructions: Find ABC.
$$a)\hspace{.2em}$$ $$A=\left[ \begin{array}{cccc}-6 & 6 & -3 & 4\\ -5 & 3 & -4 & 6\end{array}\right]$$ $$B=\left[ \begin{array}{cc}4 & -1\\ 0 & 4 \\-2 & -5\end{array}\right]$$ $$C=\left[ \begin{array}{cccc}3 & 0 & -6 & 3\\ -2 & -5 & -6 & 0\end{array}\right]$$
Instructions: Find AB.
$$b)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}-y & -6x & -2\\ x & xy & 6y\end{array}\right]$$ $$B=\left[ \begin{array}{cc}-5x & -6\\ 2 & 4 \\-3x & x\end{array}\right]$$
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#5:
Instructions: Find AB.
$$a)\hspace{.2em}$$ $$A=\left[ \begin{array}{cc}4x & xy\end{array}\right]$$ $$B=\left[ \begin{array}{ccc}x^2 & 0 & 6xy\\ -6x & 0 & -3\end{array}\right]$$
Instructions: Find ABC.
$$b)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}xy & 4x & -5x\\ x^2 & 2y & 5y\end{array}\right]$$ $$B=\left[ \begin{array}{cc}-2y & 4\\ -3x & -4 \\ -5y & y\end{array}\right]$$ $$C=\left[ \begin{array}{cc}-x & 0\\ 5xy & 5x \\ -3 & 3 \\ -2 & 2y\end{array}\right]$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cc}1 & 41\\ 22 & 20\end{array}\right]$$
$$b)$$The product AB doesn't exist since matrix A is a 2 × 3 matrix and matrix B is also a 2 × 3 matrix. For the product AB to exist, we need the number of columns in A (matrix A has 3 columns) to match the number of rows in B (matrix B has 2 rows) which is not the case.
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#2:
Solutions:
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{c}18 & -17\end{array}\right]$$
$$b)\hspace{.2em}$$ $$\left[ \begin{array}{ccc}-43 & 11 & 215\\ -25 & 11 & 109 \\ 12 & 3 & -36\end{array}\right]$$
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#3:
Solutions:
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cc}-12 & -19\\ 1 & -6 \\15 & 25\end{array}\right]$$
$$b)\hspace{.2em}$$ $$\left[ \begin{array}{ccc}-9 & -7 & 41\\ 6 & -6 & -6\end{array}\right]$$
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#4:
Solutions:
$$a)$$The product ABC doesn't exist since the product AB doesn't exist. Matrix A is a 2 × 4 matrix and matrix B is a 3 × 2 matrix. For the product AB to exist, we need the number of columns in A (matrix A has 4 columns) to match the number of rows in B (matrix B has 3 rows) which is not the case.
$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cc}5xy - 6x & -26x + 6y\\ -5x^2 - 16xy & -6x + 10xy\end{array}\right]$$
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#5:
Solutions:
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{ccc}4x^3 - 6x^2y & 0 & 24x^2y - 3xy\end{array}\right]$$
$$b)$$The product ABC doesn't exist. Matrix A is a 2 × 3 matrix and matrix B is a 3 × 2 matrix. This means that our product matrix AB is a 2 × 2 matrix. The problem comes from trying to multiply the product matrix AB by matrix C. Again, the product matrix AB is a 2 × 2 matrix, while matrix C is a 4 × 2 matrix. For the final product ABC to exist, we need the number of columns in the product matrix AB (product matrix AB has 2 columns) to match the number of rows in matrix C (matrix C has 4 rows) which is not the case.